What is the fastest way to check if a vertex is visible?

I have tried rendering the depth and retrieving the z from the x,y screen position as you can see here: Get orthographic 3d point from mouse position and depth map
But the Z isn’t completely accurate (gets more inaccurate towards the edges of the screen) and needs to convert the found depth position back into world space (which will make it accurate) and compared to the original vertex position to see if they are near each other.
In the end, it is a slower process than without doing all this…
Is this what you are talking about?